DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

4 Chapter 1. Curves2πb2πaFigure 1.6Brief review of hyperbolic trigonometric functions. Just as the circle x 2 + y 2 = 1 isparametrized by (cos θ, sin θ), the portion of the hyperbola x 2 − y 2 =1lying to the right ofthe y-axis, as shown in Figure 1.7, is parametrized by (cosh t, sinh t), wherecosh t = et + e −tand sinh t = et − e −t.22By analogy with circular trigonometry, we set tanh t = sinh t1and secht =cosh t cosh t . The(cosh t, sinh t)Figure 1.7following formulas are easy to check:cosh 2 t − sinh 2 t =1, tanh 2 t + sech 2 t =1sinh ′ (t) =cosh t, cosh ′ (t) =sinh t, tanh ′ (t) =sech 2 t, sech ′ (t) =− tanh t secht.